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In [[mathematics]], the '''Brauer–Siegel theorem''', named after [[Richard Brauer]] and [[Carl Ludwig Siegel]], is an asymptotic result on the behaviour of [[algebraic number field]]s, obtained by [[Richard Brauer]] and [[Carl Ludwig Siegel]]. It attempts to generalise the results known on the [[class number (number theory)|class number]]s of [[imaginary quadratic field]]s, to a more general sequence of number fields | |||
:<math>K_1, K_2, \ldots.\ </math> | |||
In all cases other than the rational field '''Q''' and imaginary quadratic fields, the [[regulator of a number field|regulator]] ''R''<sub>''i''</sub> of ''K''<sub>''i''</sub> must be taken into account, because ''K''<sub>i</sub> then has units of infinite order by [[Dirichlet's unit theorem]]. The quantitative hypothesis of the standard Brauer–Siegel theorem is that if ''D''<sub>''i''</sub> is the [[discriminant of an algebraic number field|discriminant]] of ''K''<sub>''i''</sub>, then | |||
: <math> \frac{[K_i : Q]}{\log|D_i|} \to 0\text{ as }i \to\infty. </math> | |||
Assuming that, and the algebraic hypothesis that ''K''<sub>''i''</sub> is a [[Galois extension]] of '''Q''', the conclusion is that | |||
: <math> \frac{ \log(h_i R_i) }{ \log\sqrt{|D_i|} } \to 1\text{ as }i \to\infty </math> | |||
where ''h''<sub>''i''</sub> is the class number of ''K''<sub>''i''</sub>. | |||
This result is [[effective results in number theory|ineffective]], as indeed was the result on quadratic fields on which it built. Effective results in the same direction were initiated in work of [[Harold Stark]] from the early 1970s. | |||
==References== | |||
* [[Richard Brauer]], ''On the Zeta-Function of Algebraic Number Fields'', ''[[American Journal of Mathematics]]'' 69 (1947), 243–250. | |||
{{DEFAULTSORT:Brauer-Siegel theorem}} | |||
[[Category:Analytic number theory]] | |||
[[Category:Theorems in algebraic number theory]] | |||
Revision as of 18:06, 25 November 2013
In mathematics, the Brauer–Siegel theorem, named after Richard Brauer and Carl Ludwig Siegel, is an asymptotic result on the behaviour of algebraic number fields, obtained by Richard Brauer and Carl Ludwig Siegel. It attempts to generalise the results known on the class numbers of imaginary quadratic fields, to a more general sequence of number fields
In all cases other than the rational field Q and imaginary quadratic fields, the regulator Ri of Ki must be taken into account, because Ki then has units of infinite order by Dirichlet's unit theorem. The quantitative hypothesis of the standard Brauer–Siegel theorem is that if Di is the discriminant of Ki, then
![{\displaystyle {\frac {[K_{i}:Q]}{\log |D_{i}|}}\to 0{\text{ as }}i\to \infty .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/373f2ccb78ea4356e20d58c1613c797e670d7645)
Assuming that, and the algebraic hypothesis that Ki is a Galois extension of Q, the conclusion is that
where hi is the class number of Ki.
This result is ineffective, as indeed was the result on quadratic fields on which it built. Effective results in the same direction were initiated in work of Harold Stark from the early 1970s.
References
- Richard Brauer, On the Zeta-Function of Algebraic Number Fields, American Journal of Mathematics 69 (1947), 243–250.